AXIOMATIC APPROACHES TO TRUTH I

Transcription

1 AXIOMATIC APPROACHES TO TRUTH I Philosophical and mathematical background Volker Halbach Axiomatic eories of Truth Neuchâtel September

2 Plan of the lecture Axiomatic theories of truth Why epistemologists and metaphysicians should be interested Why logicians may be interested Questions Classication of truth theories Disquotational theories Compositional theories Conclusion

3 Axiomatic theories of truth In an axiomatic theory of truth, the truth or satisfaction predicate is taken to be a primitive expression. Usually the term axiomatic theories of truth refers to formal axiomatic theories.

4 Axiomatic theories of truth History Tarski [] formulated and studied formal axiomatic truth theories. His adequacy condition can be understood as an axiomatization. Davidson advocated axiomatic truth theories in the s without formulating the theories. In the late s Feferman started to work on axiomatizations of Kripke s theory.

5 Why epistemologists and metaphysicians should be interested Truth and satisfaction predicates are frequently and essentially used in various areas of philosophy. Examples S knows p i S believes p, S is justied in his/her belief p and p is true (and some Gettier condition is satised). Whatever I clearly and distinctly perceive is true. ere are unknowable truths. x ( Kx Tx) ere are contingent a priori truths. Stealing money is wrong is neither true nor false.

6 Why epistemologists and metaphysicians should be interested For philosophical applications we need a truth predicate that is not relativized to a model or structure. e truth predicate is global in the sense that it is sensibly applicable to arbitrary sentences of one s language (or, perhaps, to all propositions).

7 Why epistemologists and metaphysicians should be interested Don t trust the logicians! How nonclassical logic spreads: S knows p i S believes p, S is justied in his/her belief p and p is true (and some Gettier condition is satised). I know that for any sentence the sentence itself or its negation is true. Necessarily, the truth teller is true. All sentences of the form are analytic.

8 Why epistemologists and metaphysicians should be interested A truth predicate can serve certain purposes in reductions. Adding a truth or satisfaction predicate to a language has eects that are similar to adding propositional quantiers or second-order quantiers. Hence commitment to second-order objects can be eliminated by the use of a truth or satisfaction predicate. Truth predicates can be used to express soundness claims of the sort All theorems of theory T are true.

9 Why logicians may be interested Truth predicates relativized to a model or structure are heavily used in logic. In logic, truth predicates relativized to certain language fragments are also used, e.g., partial truth predicates. A global, unrelativized truth predicate, however, is usually not denable, as Tarski s theorem on the undenability of truth shows. But one can still add such a predicate axiomatically.

10 Why logicians may be interested Proof theory As the truth predicate needs to apply to objects (sentences, propositions), the truth axioms are added to a theory of truth bearers (the base theory). e most popular choice is arithmetic. Adding truth axioms is very similar to adding second-order quantiers to arithmetic. is can help with formulating and analyzing subsystems of second-order arithmetic. Example etheoryid is equivalent to PA plus uniform disquotation for T-positive formulae: t... t n T(ṭ,...,ṭ n ) (t,...,t n )

11 Why logicians may be interested Example Systems RA <α of ramied analysis are important in the analysis of predicativity (Schütte, Feferman). ey are dicult to formulate. Using ramied truth predicates, one can give more straightforward formulations of equivalent systems. Sometimes truth theories are useful in proving theorems: Example Consider the system PA plus elementary comprehension without parameters, that is, X y (y X (y)) where (y) is pure rst-order. By interpreting this system in UTB it can easily shown to be conservative over PA.

12 Why logicians may be interested Model theory Denition A type over a model M is a nitely satised set of formulae (x, b) that have exactly the variable x free and contain at most the parameter b for one xed object b M. A type p is recursive if and only if the set of codes of formulae (x, y) with (x, b) p is recursive. Denition (recursive saturation) A model M of Peano arithmetic is recursively saturated if and only if every recursive type over M is realized (i.e. satised).

13 Why logicians may be interested CT is the system with the axioms of PA and the Tarski clauses as axioms. eorem (Kotlarski et al. [], Lachlan []) For all countable models M of PA: ere is S s.t. (M, S) CT i M is recursively saturated. e theory is conservative over PA. ere is further recent work by Enayat & Visser and Leigh.

14 . Is truth denable? Is it reducible, conservative or eliminable?. Which axioms and rules about truth can be sensibly combined?. What s the expressive and deductive power of truth? What s the role of truth in reasoning?. To what extent can theories of truth replace second-order quantiers and play a role in foundations?. How can truth be used to make explicit assumptions implicit in the acceptance of theories?. How compare dierent conceptions of truth? Are compositional truth theories always stronger than disquotational ones? How compare classical with the various nonclassical theories?. Is the theory of truth nitely axiomatizable?. Can a truth theory be categorical in some sense?. How are semantic concepts like compositionality related to mathematical concepts such as predicativity? Questions

15 Questions AXIOMATIC APPROACHES TO TRUTH II A survey of systems Volker Halbach Axiomatic eories of Truth Neuchâtel July

16 Classification of truth theories Classication of truth theories. non-classical theories ere is much work on paraconsistent and other nonclassical logics but less on full theories of truth Field [], Kremer [], Feferman [], Halbach & Horsten [], Leigh & Rathjen []. classical theories ey can be categorized along the following criteria:. typed vs type-free. disquotational vs compositional All truth theories are relative to a base theory.

17 Classification of truth theories Before we can formulate a theory of truth, we should have a theory of truth bearers, e.g., a theory of syntax, propositions, or the like. I ll use (rst-order) Peano arithmetic here. Many of the results can be be applied to other base theories, e.g. a theory of concatenation, Zermelo-Fraenkel set theory. So the truth theories are formulated in the language of rst-order arithmetic plus a unary predicate T for truth. All theories considered below are formulated in classical logic.

18 Disquotational theories Tarskian disquotation TB contains all axioms of PA plus all sentences T for all sentences without T. TB has only induction without T. is is the minimal theory that is adequate in the sense of Convention T (aer adding x (Tx Sent(x)). eories similar to TB play a role in deationary accounts of truth. eorem TB is conservative over PA and thus consistent (Tarski). e truth predicate T isn t denable in PA (undenability of truth). Any model of PA can be extended to a model of TB.

19 Disquotational theories Here are some ways to strengthen TB. Tarskian disquotation with full induction TB contains all axioms of PA including all induction axioms with T plus all sentences T for all sentences without T. eorem TB is still conservative over PA. But not any model of PA can be extended to a model of TB (Engström ).

20 Disquotational theories A disquotational theory of satisfaction uniform Tarskian disquotation with full induction UTB contains all axioms of PA including all induction axioms witht plus all sentences t... t n T(ṭ,...,ṭ n ) (t,...,t n ) where (x,...,x n ) is a formula of the language of L with exactly x,...,x n free. eorem UTB is conservative over PA. Typed disquotational theories are weak (conservative over PA). Also, they don t prove generalizations such as: x y Sent(x. y) (T(x. y) T(x) T(y))

21 Disquotational theories Untyped disquotational theories can be very strong. eorem Any theory extending PA can reaxiomatized by a set of disquotation sentences T over PA (McGee [] using a variant of Curry s paradox). But these theories are not well motivated.

22 Disquotational theories uniform Tarski disquotation for T-positive sentences PUTB contains all axioms of PA including all induction axioms witht plus all sentences t... t n T(ṭ,...,ṭ n ) (t,...,t n ) (x,...,x n ) is a formula of the language with T with exactly x,...,x n free such that T does not occur in the scope of ( and are the only connectives). eorem PUTB is equivalent to RA <є and KF below.

23 Disquotational theories ere are reasonable, stronger disquotational theories. For instance, one can get second-order arithmetic (without second-order parameters) from a disquotational truth theory (Schindler). Generally, disquotational theories can vary signicantly in their properties, depending on how the paradoxes are blocked.

24 Compositional theories All knownnatural disquotational theories fail to prove generalizations such as x y Sent(x. y) (T(x. y) T(x) T(y)) us various philosophers have tried to add them as axioms, although Tarski was dismissive of such attempts. Compositional axioms may allow nite axiomatizability.

25 Compositional theories typed compositional truth e system CT is given by all the axioms of PA and the following axioms: s t T(s=. t) s = t x Sent(x) (T(. x) Tx) x y Sent(x. y) (T(x. y) T(x) T(y)) x y Sent(x. y) (T(x. y) T(x) T(y)) v x Sent(. vx) (T(. vx) tt(x(tv))) v x Sent(. vx) (T(. vx) tt(x(tv))) Induction is restricted to sentences without T. eorem ([],[],[]) CT is conservative over PA.

26 Compositional theories ere are claims in the literature to the eect that the Tarskian clauses x the extension of the truth predicate. If a model of PA can be expanded to a model of CT at all (in which case it will be rec. saturated), there will be uncountably many ways to do so. Also CT does not prove hat all sentences of the form are true. So let s add induction. = = =... =

27 Compositional theories typed compositional truth with full induction CT is CT plus all induction axioms in the language with T. Using induction with T one proves in CT that all axioms of PA are true, and then using induction again, one proves that all theorems of PA are true. Since = is not true, CT proves that PA is consistent. eorem CT fails to be conservative over PA. e eect of adding the CT axioms to PA is the same as adding elementary comprehension. CT is equivalent to ACA.

28 Compositional theories No truth theory containing TB is categorical (Beth s theorem). But CT xes the extension of the truth predicate in the following way: eorem Let CT be CT with the predicate T instead of T. etheory CT CT plus induction in the mixed language proves x Sent(x) (Tx T x).

29 Compositional theories To get even stronger theories one can iterate the theory CT along some ordinal notation system. e system turns out to be equivalent with iterated predicative comprehension.

30 Compositional theories Type-free compositional theories are usually extensions of CT. Criteria for choosing theories:. consistency, ω-consistency, and absence of non-trivial models. deductive and conceptual power. transparency or symmetry. Leitgeb: What theories of truth should be like (but cannot be) All theories below have full induction.

31 Compositional theories FS is a system of classical and symmetric truth. Friedman Sheard e system FS has all axioms of PA plus the following axioms and rules: s t T(s=. t) s = t x Sent T (x) (T. x Tx) x y Sent T (x. y) (T(x. y) (Tx Ty)) x y Sent T (x. y) (T(x. y) (Tx Ty)) v x Sent T (. vx) (T(. vx) tt(x(tv))) v x Sent T (. vx) (T(. vx) tt(x(tv))) T T

32 Compositional theories eorem e liar sentence is neither provable nor refutable in FS. is not needed for proof-theoretic strength; can be expressed via iterated reection. Natural models of FS can be obtained via nitely iterated revision (in Gupta Herzberger style). FS is ω-inconsistent. (McGee []) FS is equivalent to nitely iterated Tarskian truth, i.e., iterated CT.

33 Compositional theories is is an axiomatization of Kripke s[] theory of truth with Strong Kleene logic. Kripke Feferman e system KF is given by all the axioms of PA and the following axioms: s t T(s=. t) s = t s t T(. s=. t) s = t x Sent T (x) (T(.. x) Tx) x y Sent T (x. y) (T(x. y) Tx Ty) x y Sent T (x. y) (T. (x. y) T. x T. y) x y Sent T (x. y) (T(x. y) Tx Ty) x y Sent T (x. y) (T. (x. y) T. x T. y)

34 Compositional theories v x Sent T (. vx) (T(. vx) tt(x(tv))) v x Sent T (. vx) (T(.. vx) tt(. x(tv))) v x Sent T (. vx) (T(. vx) tt(x(tv))) v x Sent T (. vx) (T(.. vx) tt(. x(tv))) t (T(Ṭt) Tt ) t T. Ṭt (T. t Sent T (t )) x Sent T (x) (Tx T. x) Several variants of KF can be found in the literature. is is my version. e last axiom called the consistency axiom excludes truth-value gluts.

35 Compositional theories eorem KF is an axiomatization of Kripke s theory of truth with the Strong Kleene schema. It s an axiomatization of a partial notion of truth in classical logic. KF proves the liar sentence, as it excludes truth-value gluts. Even without the consistency axiom, KF cannot be closed consistently under and. It s essentially asymmetric. KF is equivalent to є = ω ω ω iterated Tarskian truth (Feferman []). Feferman s analysis proceeds in terms of innite conjunctions.

36 Compositional theories Further systems: Variations of KF: Feferman s [] strong reective closure of PA, weak Kleene, Feferman s [] DT Cantini s [] VF and supervaluations axiomatizations of stable truth?

37 Some preliminary conclusions Conclusion e metamathematical property of the axioms systems diers signicantly, depending on which solution of the paradoxes is adopted. Typed disquotational theories are proof-theoretically weak (conservative); type-free disquotational theories can be very strong. So, don t assume that disquotation is weak. Typed truth theories give at most elementary comprehension. Compositional truth theories don t seem to exceed the strength of predicative systems. Strong type-free truth predicates tend to be partial (or paraconsistent ); but presenting these truth predicates in a nonclassical logic will weaken them.

38 Conclusion Conclusions for the logician: Axiomatic theories of truth expand the playground. for the philosopher: We have various truth theories on oer. But they dier wildly in their properties. Pick your favourite!

39 Conclusion Andrea Cantini. AtheoryofformaltrutharithmeticallyequivalenttoID. Journal of Symbolic Logic, :,. Ali Enayat and Albert Visser. Full satisfaction classes in a general setting (part I). Technical report,? to appear in Logic Group Preprint Series,availableathttp:// Solomon Feferman. Towards useful type-free theories I. Journal of Symbolic Logic, :,. Solomon Feferman. Reflecting on incompleteness. Journal of Symbolic Logic, :,. Solomon Feferman. Axioms for determinateness and truth. Review of Symbolic Logic, :,. Hartry Field. Saving Truth From Paradox. Oxford University Press, Oxford,. Volker Halbach and Leon Horsten. Axiomatizing Kripke s theory of truth. Journal of Symbolic Logic, :,. Henryk Kotlarski, Stanislav Krajewski, and Alistair Lachlan. Construction of satisfaction classes for nonstandard models. Canadian Mathematical Bulletin, :,. Michael Kremer. Kripke and the logic of truth.

40 Conclusion Journal of Philosophical Logic, :,. Saul Kripke. Outline of a theory of truth. Journal of Philosophy, :,. reprinted in []. Alistair Lachlan. Full satisfaction classes and recursive saturation. Canadian Mathematical Bulletin, :,. Graham Leigh and Michael Rathjen. The Friedman Sheard programme in intuitionistic logic. Journal of Symbolic Logic, :,. Graham E. Leigh. Conservativity for theories of compositional truth via cut elimination.. [math.lo]; Robert L. Martin, editor. Recent Essays on Truth and the Liar Paradox. Clarendon Press and Oxford University Press, Oxford and New York,. Vann McGee. How truthlike can a predicate be? A negative result. Journal of Philosophical Logic, :,. Vann McGee. Maximal consistent sets of instances of Tarski s schema (T). Journal of Philosophical Logic, :,. Alfred Tarski. Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica Commentarii Societatis Philosophicae Polonorum, :,.

41 Conclusion reprinted as The Concept of Truth in Formalized Languages in [, ]; page references are given for the translation. Alfred Tarski. Logic, Semantics, Metamathematics. Clarendon Press, Oxford,.